We propose to extend the scope of "shake-and-bake" algorithms for direct methods of phase determination by developing new probabilistic theory to fully exploit conditional information available in SIR (single- derivative isomorphous replacement), SAS (single-wavelength anomalous scattering), and SIRAS cases. We expect that the new theory will provide improved tangent formulas and minimal principles, which will be able to solve the phase problem for a macromolecular crystal using only one, single-wavelength data set, even in the common case that the diffraction data do not extend to atomic resolution. One focus of the new theoretical derivations will be the use of a known substructure of heavy resolution. One focus of the new theoretical derivations will be the cause of a known substructure of heavy or anomalously scattering atoms, e.g., the case of a known selenium substructure of a selenomethionine protein. Such cases are natural extensions of the by now rather numerous determinations of multi-selenium substructures using appropriately renormalized SAS Friedel-or Bijvoet-pair differences data in the SnB computer program. Second theoretical focus will be derivation of enantiomorph-specific probabilistic theory of structure invariants has been measured by recording Renninger three-beam interference effects. We also propose to develop new methods for phasing low-resolution data from crystals of large biomolecules and biomolecular complexes. Our goals are new methods for ab initio phase estimation using randomly positioned (polyatomic) globs as starting models, and for subsequent model-free phase refinement and extension to higher resolution. This research will develop new reciprocal-space omit-map correlation - coefficient techniques to rank multiple sets of trial phases and to validate phase extension, and it will seek ways to exploit algebraic (as distinct from approximate probabilistic) phase relationships for one, two-, and three-atom P1 structures to model macromolecular structures at very low resolution.